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Magnetic flux tube
A magnetic flux tube is standing vertically out of the solar photosphere. The photospheric gas pressure outside the tube at the bottom of the photosphere (z = 0 km)
is Pe (z = 0) = 104 Pa. Inside the flux tube the gas pressure is only 10% of the gas
pressure outside the tube. Assume the the solar atmosphere is isothermal with a
temperature of T = 6000 K and a mean molecular weight µ = 0.6.
(a) Calculate the magnetic field strength at the bottom of the photosphere inside
the tube, assuming the tube to be in hydrostatic equilibrium.
(b) Assume that the gas outside the flux tube is in hydrostatic equilibrium, and
deduce a expression for the pressure as a function of height
Hint: Write down the equation for hydrostatic equilibrium, and replace the
density with an expression for the density from the equation of state, to get a
differential equation for the pressure and solve it.
(c) In order for the flux tube to remain in pressure equilibrium, it must expand
with height. Calculate the diameter of the tube at the top of the chromosphere
(z=1500 km), assuming that the tube remains circular and that the gas pressure
has the same pressure scale height inside as it has outside the flux tube. Assume
that the magnetic flux tube is circular with a diameter of d=100 km at z=0 km.
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